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Chapter 2: Problem 54

State the additive inverse of the given integer. 10

Short Answer

Expert verified
The additive inverse of 10 is -10.

Step by step solution

01

Understanding Additive Inverse

The additive inverse of a number is a number that, when added to the original number, results in zero. Thus, the additive inverse of any number \( x \) is \(-x\).
02

Apply the Concept to the Exercise

Identify the given integer, which is 10. According to the concept of additive inverse, we need to find a number \( x \) such that \( 10 + x = 0 \). This implies \( x = -10 \).
03

Verify the Additive Inverse

Add 10 and -10 together to ensure they sum to zero: \( 10 + (-10) = 0 \). This confirms that -10 is indeed the additive inverse of 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Integers
Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. They do not include fractions or decimals. Integers extend infinitely in both the positive and negative directions on the number line, represented as \(..., -3, -2, -1, 0, 1, 2, 3, ...\).

These are important for performing various mathematical operations. Positive integers are numbers greater than zero. Negative integers, on the other hand, are less than zero. Zero itself is an important integer as it acts as a neutral element in addition. Knowing about integers helps understand many basic and advanced math operations.
Exploring Negative Numbers
Negative numbers are all the integers less than zero. They are placed on the left side of zero on the number line. These numbers are essential in denoting values less than zero, such as temperatures below freezing or a monetary debt.

Negative numbers have distinct properties when it comes to arithmetic operations:
  • When added to a positive number, the larger absolute value determines the result's sign.
  • Adding a negative number effectively decreases value.
  • Subtracting a negative number, however, makes it a positive addition.
Being adept at manipulating negative numbers is vital for solving equations and real-world problems.
Basic Arithmetic Operations
Basic arithmetic operations include addition, subtraction, multiplication, and division. Each operation has specific rules, especially when dealing with positive and negative numbers.

Let's revisit some fundamental rules:
  • Addition: When adding two numbers of opposite signs, subtract their absolute values and take the sign of the larger number.
  • Subtraction: Subtracting is essentially adding the additive inverse of a number. For instance, subtracting \(b\) is the same as adding \(-b\).
  • Multiplication and Division: When the signs of the numbers are the same, the result is positive; when different, the result is negative.
Understanding these operations ensures smooth problem-solving, especially when working with integers and the concept of additive inverses.

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